Math on science (I)

Ever wondered how much influence math has in our everyday life?

It is not too much to say that almost EVERYTHING surrounding us is related to math, from the most obvious (engineering, statistics, computers etc) to less intuitive areas such as music and even biology.

As Galileo Galilei said in 1623: "The book of nature is written in the language of mathematics".

I'm sure all of you have heard the name Leonardo Da Vinci, some his contributions to so many areas in science, arts etc. are world famous, but maybe the name Leonardo De Pisa is not so well known. At least by his first name but, what about Fibonacci?

Fibonacci

Leonardo Pisano (1170 -1250) was a merchant and mathematician who lived in Italy and became known for spreading the hindu-arabic numeration system in Europe through his book Liber Abaci (Book of Calculation). In this book, as a solution for the problem of how the population of rabbits grows, it was written for the first time the sequence of numbers now known as Fibonacci Numbers:

1, 1, 2, 3, 5, 8, 13, 21, 34...

Notice any pattern? In Fibonacci's Sequence you get every number by adding the two previous ones.

It may seem that there is nothing else to this sequence than meets the eye, and in fact it would have been like that if biologists and other scientists hadn't found some interesting properties.

The way that the shell of snails grow can relates to the fibonacci sequence in the way you can see in the following picture:

 As you can see, the side of each square is a number of the sequence.
There is also an interesting relation between these numbers and the way plants grow. Plants try to be as efficient as possible when absorbing light from the Sun, son no leaf is over another one. Usually branches in plants form a spiral around the stem. If you take the leaf at the bottom of the stem and count it as 0, and then you count all the leaves from there to the top, for most plants you will get a number of the Fibonacci sequence. Also if you count the laps that took you to get to the top (upwards spiral) it will also be a term of the sequence.
The way seeds are arranged in daisies and pine cones also form spirals that go around a number of times that is a Fibonacci number. If you count the petals in each layer in roses, you will alaso notice something...

Also the phalanxes in your fingers follow this rule. You have to admit there is a lot to this sequence....but there's more...
In a bigger scale, the arms of the spirals that galaxies form, are also arranged according to Fibonnaci's numbers.

But the last property I want to talk about today is one that will deserve a post on its own in the future: The Golden Ratio.

The golden ratio is an irrational number represented by the greek letter phi (φ or Φ = 1.6180339...), and it has been around for thousands of years. For artists and architects, it's a proportion that represents beauty. There  is a lot to say regarding the golden proportion, but as I mentioned before, we'll do that in the future. For now I want to talk to you about the relationship between this number and Fibonacci. As it turns out, if you divide any term of the sequence by its predecessor, the further you go into the sequence, the closer you will get to the golden ratio.

It is certainly curious that this sequence has so many properties, there are obviously more that the ones I mentioned, some of them related to advanced math, but I find it amazing that when Fibonacci thought of it, he had no idea of all it would mean...

Find your limits

I'm sure by now most of you are more than familiar with the concept of limit of a function, but I believe it is a great place to start our blog with.

At first, the idea of limit of a function in a point was used by mathematicians without worrying of having a precise definition for it. When the time came for mathematical analysis principles to be formalized, it was A.L. Cauchy (1789-1857) who wrote the definitive version, the one we still use today. It goes like this:

A function f(x) has a limit L in a point p if for every real number ε > 0 exists another real number δ > 0 such as |f(x) - L| < ε if 0 < |x - p| < δ and x ≠ p, where δ is chosen depending on ε. It is written

But, what does this exactly mean? What this definition tells us is that when x takes values close to p, as close as you want, but without being equal to p, both from the left and the right of p (higher and lower values than p), the function approaches to a value L, which is the limit of the function, as much as we want.

With this definition it comes natural to assume that for the limit L to exist, it's value must be the same whether we approach from the left or from the right. In fact, this is the only requirement for the limit to exist, as we don't need the function to exist in a certain point for it to have a limit in that point. We will understand this better with an example:

We want to find the limit of $$\displaystyle\lim_{x \to{1}}{\displaystyle\frac{x-1}{x^2-1}}$$

If we try replacing the x by 1 we will get the following: 
$$ \frac{x-1}{x^{2}-1} = \frac{1-1}{1^{2}-1} = \frac{0}{0} $$
This is what we call an indetermitation, which means that the function does not have a representation in that point, ie, it does not exist.

In this case we have that f(1) does not exist, but we can still try to find the limit. If we take a closer look at how the function behaves when approaching 1 from right and left, it's relatively easy to see where things are going.

f(1,5)= 0,4...                                           f(0,5)= 0,66...
f(1,25)= 0,44...                                       f(0,75)= 0,57...
f(1,1)= 0,47...                                         f(0,9)= 0,52...
f(1,01)= 0,49...                                       f(0,99)= 0,502...

Right now it seems intuitive to think that if there was a f(1), it would be equal to 0,5. 
So, how can we solve this limit without having to make so many tedious calculations? Well, there are several tools, techniques and methods you can use to solve limits, but usually the simplest solution is the best one, let's see 
$$ \frac{x-1}{x^{2}-1} = \frac{x-1}{\left(x-1\right) \cdot \left(x+1\right)} = \frac{1}{x+1} = \frac{1}{2} $$
By cancelling out the (x-1) we get the solution to the limit.

This is problably the easiest method and the easiest type of limit, but math, as building houses, always has to start with the foundations. There will me more math to come.....stay tuned!!

Aula UE

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Welcome!

This blog has been created by Jaime Manuel Martínez Pérez as an integrated project for the 1st year Aerospace Engineering Degree taught at the Polytechnic School of the “Universidad Europea de Madrid”. Academic Year 2013-2014.

The goal of this blog is that we all learn a little science from each other and, why not, have some fun while we're at it. I'm really looking forward to reading your comments, so please, feel free to do so, whether it is to discuss, propose a new subject...whatever!!